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Have the mechanic hook a vacuum gauge to the intake manifold.
Start the engine and put something between the ISC(idle speed control) motor plunger tip and throttle linkage pad to keep the throttle open to where it will keep running at a steady RPM.

on the bottom rear driver side of the cfi there is a gold metal vac line fitting.

flip the cfi over and if that area if full of carbon build up, you will have the problem your speaking of.~yet you said the cfi was recently rebuit.

in my diy link there is a lot of cfi tips but.....

first,,,,

do you have fuel presure at the shrader= YES

do you have "whiteish blue spark"??????????

is the compression good enough across all cylinders.??????

since the car sat all winter,, eliminate bad gas by doing this (in diy link)....
-jumper the tan / light green wire to the frame of the car at the triangle shaped code pulling connector in the engine bay up by the master cylinder.
-disconnect the fuel supply line at the rear of the CFI
-rig up a hose on the end like a garden hose or something that about 5' long.
-drop the hose into a container to catch fuel.
-turn the key "FORWARD ONLY"
-the fuel pump will run 100% of the time now and extract all the fuel from the tank.
-when fuel is done dumping, turn off key and button things up then remove jumper wire.
-add new fuel
-turn key on for 5 seconds and turn key off , repeat this 3 times to "refill" your fuel lines and fuel system
bad fuel may be the reason why when you introduce "different fuel" the car runs

------------------------------------------------------------
on this car,, to answer your earlier question..................
you should never need to push on the gas to start... if its pushed to the floor, the fuel injectors actually shut off.
-in the trunk there is a fuel pump relay located "most of the time" outboard of the passanger side trunk hinge.
-there is a safety switch if rear ended, located "sometimes" inside the trunk on the rear driver side, black box with a white button.
-this safety switch removes power delivery from the fuel pump
-if you push the button down, that allows power to reach the fuel pump.

-------------------------------------- just notes / fyi for your future information..............
if the fuel pump is not powering up.....
-go to the fuel pump relay in the trunk
-there will be a small tan/light green wire.
-side tap and ground the tan/ light green wire and ground it to the frame of your car.
-if car starts,, let us know.

i say yes because you say you replaced the pick up ,, which is down inside the distributor,, and ....so..............

did they dial in top dead center perfectly and drop in the distributor or is is possibly off one tooth??

i know from personal experience that the 3.8L cfi **WILL NOT** tolerate being off one tooth,, no matter if the tech says "well i should be able to adjust one way or the other for that".,,, nope,, noda. being off one tooth may not cause a backfire but *will* cause the symptom you are having.

Have the mechanic hook a vacuum gauge to the intake manifold.
Start the engine and put something between the ISC(idle speed control) motor plunger tip and throttle linkage pad to keep the throttle open to where it will keep running at a steady RPM.

i say yes because you say you replaced the pick up ,, which is down inside the distributor,, and ....so..............

did they dial in top dead center perfectly and drop in the distributor or is is possibly off one tooth??

i know from personal experience that the 3.8L cfi **WILL NOT** tolerate being off one tooth,, no matter if the tech says "well i should be able to adjust one way or the other for that".,,, nope,, noda. being off one tooth may not cause a backfire but *will* cause the symptom you are having.

Thanks jcassity. I will deliver your ideas to the mechanic as well. Sorry for not responding sooner but I've been busy on other projects. Hopefully one of your solutions will lead to fixing this. I miss my Cougar.

Another thing came to mind, the 3.8L CFI cars use a time delay relay for the EEC power.
The relay takes about 5-10 seconds to drop out after you turn the key off.
If you listen real close to the center area of the dash, you can hear it click when it drops.

This gives the EEC time to go through a shutdown routine before it turns off.
The EEC will extend the ISC motor plunger to a "fast idle" position to prepare for the next start-up. (The engine has already stopped)
Don't know if it does anything else.

If someone put a standard relay in the EEC power relay socket, you will lose this shut-down procedure.

So it's now August 15 and the car still isn't getting enough fuel to the new injectors to make the car idle. Could the solution be to replace the entire distributor? The distributor in the car now is the original. Any thoughts??